| Project/Area Number |
23KF0051
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Multi-year Fund |
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| Section | 外国 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kobe University |
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Principal Investigator |
Rossman W.F 神戸大学, 理学研究科, 教授 (50284485)
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| Co-Investigator(Kenkyū-buntansha) |
RAUJOUAN THOMAS 神戸大学, 理学研究科, 外国人特別研究員
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| Project Period (FY) |
2023-04-25 – 2025-03-31
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| Project Status |
Granted (Fiscal Year 2023)
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| Budget Amount *help |
¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2024: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2023: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Keywords | 曲面理論 / 可積分系 / Darboux変換 / DPW方法 |
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| Outline of Research at the Start |
The DPW method will be the primary tool in this research, both for smooth surfaces and for discrete surfaces. Topological questions will be considered, and applied to create surfaces with nontrivial topologies. For such complicated surfaces, symmetry properties and embeddedness will then be considered, as described below.
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| Outline of Annual Research Achievements |
We aim at constructing examples of surfaces with constant mean curvature in various ambient spaces using integrable system techniques in the context of holomorphic maps. These techniques find their origins in the Weierstrass representation (1866), which have been extended by DPW (1998) and are now one of the main tools for this task. They extend to the construction of discrete analogues of smooth surfaces.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
We have investigated Delaunay ends for constant mean curvature (CMC) surfaces in Euclidean and hyperbolic space when constructed via the DPW method. We developed a method to check wether a surface arising from DPW has self-intersections, and constructed new examples of complete, embedded, CMC surfaces with any number of Delaunay ends in the hyperbolic space.
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| Strategy for Future Research Activity |
1) With N. Schmitt and J. Ziefle: we have been translating the Weierstrass and Bryant reprensentations for minimal surfaces into a gauge theoretic framework. This allows for the construction of catenoidal ends arising from Fuchsian systems, and a dressing action on the holomorphic frame induces what should be a Darboux transformation.
2) With L. Heller, we are constructing new examples of minimal surfaces in the three-dimensional sphere which have high genus and are not Lawson surfaces. We will obtain surfaces constructed by Kapouleas and should be able to compute their area, as their genus goes to infinity.
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